A note on the partially truncated Euler-Maruyama method

The partially truncated Euler-Maruyama (EM) method was recently proposed in our earlier paper [3] for highly nonlinear stochastic differential equations (SDEs), where the finite-time strong Lr-convergence theory was established. In this note, we will point out that one condition imposed there is restrictive in the sense that this condition might force the stepsize to be so small that the partially truncated EM method would be inapplicable. In this note, we will remove this restrictive condition but still be able to establish the finite-time strong Lr-convergence rate. The advantages of our new results will be highlighted by the comparisons with our earlier results in [3].